Submodularity is an important  property of set functions with deep theoretical results  and various  applications. Submodular systems appear in many applicable area, for example machine learning, economics, computer vision, social science, game theory and combinatorial optimization.  Nowadays submodular functions optimization has been attracted by many researchers.  Pendant pairs of a symmetric...

We introduce three new cut tree structures of graphs G in which the vertex set of the tree is a partition of V (G) and contractions of tree vertices satisfy sparsification requirements that preserve various types of cuts. Recently, Kawarabayashi and Thorup [8] presented the first deterministic near-linear edge-connectivity recognition algorithm. A crucial step in this algorithm uses the existen...

In the title compound, C(24)H(28)N(2)O(4)S(2), the dihedral angles between the central benzene ring and the pendant rings are 77.44 (11) and 79.23 (10)°, and the dihedral angle between the pendant rings is 23.31 (12)°. Both sulfonamide groups project to the same side of the central benzene ring and the mol-ecule has approximate non-crystallographic mirror symmetry. One of the ethyl side chains ...

Reaction of [Tb(III)(DMSO)(8)](CF(3)SO(3))(3) with a resorcinarene cavitand substituted by four -CH(2)P(O)Ph(2) pendant arms afforded an infinite 1D inorganic polymer, in which the consecutive resorcinarene units are linked through two terbium strands, this connection involves pairs of phosphoryl groups occupying adjacent positions of the resorcinarene units.

In the title compound, C26H16ClN3, the dihedral angles between the central pyridine ring and the pendant phenyl, chloro-benzene and indole rings are 18.52 (12), 48.97 (11) and 21.20 (10)°, respectively. An intra-molecular C-H⋯Nc (c = cyanide) hydrogen bond occurs. In the crystal, inversion dimers linked by pairs of N-H⋯Nc hydrogen bonds generate R 2 (2)(16) loops.

Photophysical investigations are reported for two arene-derivatized zirconium phosphonates (layered solids). The chromophore groups that are attached within the interlayer region of these materials are seen to form excimer pairs. Whereas the naphthalene-containing system exhibits both monomer and excimer fluorescence, the potentially greater overlap of chromophores for the system containing ant...

Let G be a graph with minimum degree δ. It is well-known that maximal adjacency orderings (MAOs) compute a vertex set S such that every pair of S is connected by at least δ internally vertex-disjoint paths in G. We present an algorithm that, given any pair of S, computes these δ paths in linear time O(n+m). This improves the previously best solutions for these special vertex pairs, which were f...

In the title compound, C23H18O4, the meth-oxy-benzene ring and attached C=C grouping are disordered over two sets of sites in a 0.823 (5):0.177 (5) ratio. The dihedral angles between the central benzene ring and the pendant phenyl and meth-oxy-benzene ring (major orientation) are 51.21 (1) and 51.6 (1)°, respectively. In the crystal, inversion dimers linked by pairs of C-H⋯O hydrogen bonds gene...

In the title compound C20H13N5, the dihedral angle between the carbazole ring system (r.m.s. deviation = 0.027 Å) and the pendant benzene ring is 55.08 (6)°. One of the azide N atoms is disordered over two positions in a 0.65 (2):0.35 (2) ratio. In the crystal, aromatic π-π stacking is observed [minimum centroid-centroid separation = 3.6499 (13) Å] as well as inversion-dimers connected by pairs...

In the title compound, C28H25N3O4, the central pyrrolidine ring adopts adopts an envelope conformation with the N atom as the flap and the piperidine ring adopts a chair conformation. The pendant pyrrolidine ring is almost planar (r.m.s. deviation = 0.008 Å). An intra-molecular C-H⋯O inter-action closes an S(6) ring. In the crystal, inversion dimers linked by pairs of N-H⋯O hydrogen bonds gener...